Download Problem of the Week

Problem of the Week
Problem C and Solutions
Pesky Products
Problem
A sequence contains three positive numbers such that:
(i) the product of the first and second numbers equals the third number;
(ii) the product of the second and third numbers is 180; and
(iii) the second number is five times the third number.
Determine the product of the three numbers in the sequence.
Solution 1
In this solution we will try to find the numbers by working with the factors of 180.
The product of the second and third numbers is 180 and the second number is five times the
third number. The number 180 can be written as 2 × 2 × 3 × 3 × 5. By playing with the factors
we can get the second number 5 × 2 × 3 and the third number 2 × 3. That is, the second
number could be 30 and the third number could be 6.
Now using the fact that the first number times the second number is equal to the third
number, we see that some number times 30 equals 6 and it follows that the first number would
be 6 ÷ 30 = 15 = 0.2.
The product of the three numbers is
1
5
× 30 × 6 = 6 × 6 = 36.
This solution only works because the second and third numbers in the sequence happen to be
integers.
Solution 2
Let the three numbers be represented by a, b, and c.
Since the product of the first and second numbers equals the third number, a × b = c. We are
looking for a × b × c = (a × b) × c = (c) × c = c2 . So when we find c2 we have found the
required product a × b × c.
We know that b × c = 180 and b = 5 × c, so b × c = 180 becomes (5 × c) × c = 180 or
5 × c2 = 180. Dividing by 5, we obtain c2 = 36. This is exactly what we are looking for since
a × b × c = c2 .
Therefore, the product of the three numbers is 36.
For those who need to know what the actual numbers are, we can proceed and find the three
numbers. We know c2 = 36, so c = 6 since c is a positive number. So b = 5 × c = 5 × (6) = 30.
6
And finally, a × b = c so a × (30) = 6. Dividing by 30, we get a = 30
= 15 = 0.2. We can verify
the product a × b × c = (0.2) × (30) × (6) = 6 × 6 = 36.